I am trying to compute the cohomology in Hartshorens's exercise 4.7 in chapter3. Given a subscheme $Y$ in $X=P^2_k$ defined by a homogeneous equations $f(x_0,x_1,x_2)$ of degree $d$, and point (1,0,0) is not contained in $Y$. I want to compute its Cech cohomology. And the following is what I attempt so far.
First, according to the hint, we can cover $Y$ with two affine subset $U=X \cap ${$x_1 \neq0$} and $V=X \cap${$x_2\neq0$}. Then, we give the sheaf of regular functions $\mathcal{O}_Y$ on $Y$. On $U$($V$, resp.), that is $\Gamma(U, \mathcal{O}_U)=k[\frac{x_0}{x_1},\frac{x_2}{x_1}]/(f)$ (and $\Gamma(U, \mathcal{O}_V)=k[\frac{x_0}{x_2},\frac{x_1}{x_2}]/(f)$, resp.).
And we define $d$: $\Gamma(U,\mathcal{O}_U)\oplus\Gamma(V,\mathcal{O}_V)\to \Gamma(U\cap V, \mathcal{O}_{U\cap V})$ simply by addition. That is
$<g[\frac{x_0}{x_1},\frac{x_2}{x_1}]/(f), h[\frac{x_0}{x_2},\frac{x_1}{x_2}]/(f)>\to (g+h)/(f)$ in $<k[\frac{x_0}{x_1},\frac{x_2}{x_1},\frac{x_0}{x_2},\frac{x_1}{x_2}]/(f)>$
Now, $H^0=ker$ $d$, and this can be done only when $g,h\in (f)$, so $dimH^0=1$; to compute $H^1$, we notice that the cokernel is of the form {$\frac{x_0^i}{x_1^j x_2^k}:j+k=i$}. But I got stuck in this step, and I don't know how to use the condition of $f$ to determine the expilict form of $H^1(Y, \mathcal{O}_Y)$.
Am I right so far? Hope someone could help. Thanks!